If you have only 6 hours from touchdown to takeoff, it's going to be really tight. The Maglev train is quick (maybe 15 minutes) to Longyang station, but you could easily wait 15-20 minutes for the next train. (As an aside, if you show your same day arrival/departure air tickets, you get a discount on the maglev.) You could take a taxi from Longyang, or the Shanghai Metro can more quickly take you to where you want to go (surface traffic is almost always awful). Also, the metro is significantly cheaper than a taxi (maybe 3-5 yuan instead of 30-50, though still quite cheap after converting to NZD!).
From Longyang, metro line 2 will take you directly to Lujiazui (Pudong business area) where there are many tall buildings and you can get a taste for "new" Shanghai. If you go a little further (East Nanjing) and then transfer to line 10 to Yuyuan Garden, you can see a touristy part of "old" Shanghai. I put "new" and "old" in quotes because you can really find new and old intermixed throughout the city. If you can manage the schedule, I would probably suggest Yuyuan over Pudong.
I would allow at least an hour to navigate the Maglev and metro to Yuyuan station, for example. That would leave you maybe an hour or two if you're lucky to explore your destination.
Finally, check your PVG-AKL flight status when you arrive, as both times I've flown with Air NZ on that leg the flight has been considerably delayed (5 to 12 hours). You may have a lot more time than planned!